## Abstract

The motivation for the research reported in this paper comes from modeling the spread of vector-borne virus diseases. To study the role of the host versus vector dynamics and their interaction we use the susceptible-infected-removed (SIR) host model and the susceptible-infected (SI) vector model. When the vector dynamical processes occur at a faster scale than those in the host-epidemics dynamics, we can use a time-scale argument to reduce the dimension of the model. This is often implemented as a quasi steady-state assumption (qssa) where the slow varying variable is set at equilibrium and an ode equation is replaced by an algebraic equation. Singular perturbation theory will appear to be a useful tool to perform this derivation. An asymptotic expansion in the small parameter that represents the ratio of the two time scales for the dynamics of the host and vector is obtained using an invariant manifold equation. In the case of a susceptible-infected-susceptible (SIS) host model this algebraic equation is a hyperbolic relationship modeling a saturated incidence rate. This is similar to the Holling type II functional response (Ecology) and the Michaelis-Menten kinetics (Biochemistry). We calculate the value for the force of infection leading to an endemic situation by performing a bifurcation analysis. The effect of seasonality is studied where the force of infection changes sinusoidally to model the annual fluctuations of the vector population. The resulting non-autonomous system is studied in the same way as the autonomous system using bifurcation analysis.

Original language | English |
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Pages (from-to) | 4314-4338 |

Number of pages | 25 |

Journal | Mathematical Biosciences and Engineering |

Volume | 16 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

### Funding

This publication is based upon work from COST Action CA16227 Investigation & Mathematical Analysis of Avant-garde Disease Control via Mosquito Nano-Tech-Repellents, supported by COST (European Cooperation in Science and Technology). Weblink www.cost.eu. Peter Rashkov acknowl- edges financial support from the Bulgarian National Fund for Scientific Research (FNI) under contract DKOST01/29 and would like to thank the Mathematical Biosciences Institute (funded from the National Science Foundation Division of Mathematical Sciences and supported by The Ohio State University) for the helpful discussions during the Emphasis Semester on Infectious Diseases: Data, Modeling, Decisions - Spring 2018. Maira Aguiar has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 792494. Nico Stollenwerk is supported by National Funding from FCT - Fundac¸ão para a Ciência e a Tecnologia, under the project: UID/MAT/04561/2019, and supported as an Investigador FCT.

Funders | Funder number |
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Fundac¸ão para a Ciência e a Tecnologia | UID/MAT/04561/2019 |

Marie Skłodowska-Curie | |

Mathematical Biosciences Institute | |

Division of Mathematical Sciences | |

Ohio State University | |

Field Neurosciences Institute | DKOST01/29 |

Horizon 2020 Framework Programme | 792494 |

European Cooperation in Science and Technology | |

Fundação para a Ciência e a Tecnologia | |

Fonds De La Recherche Scientifique - FNRS |

## Keywords

- Asymptotic expansions
- Epidemic models
- Geometrical singular perturbation
- Quasi steady state assumption
- Seasonally-forced models
- Vector borne disease dynamics